An iterative row-action method for interval convex programming

  • Y. Censor
  • A. Lent
Contributed Papers

Abstract

The iterative primal-dual method of Bregman for solving linearly constrained convex programming problems, which utilizes nonorthogonal projections onto hyperplanes, is represented in a compact form, and a complete proof of convergence is given for an almost cyclic control of the method. Based on this, a new algorithm for solving interval convex programming problems, i.e., problems of the form minf(x), subject to γ≤Ax≤δ, is proposed. For a certain family of functionsf(x), which includes the norm ∥x∥ and thex logx entropy function, convergence is proved. The present row-action method is particularly suitable for handling problems in which the matrixA is large (or huge) and sparse.

Key Words

Interval convex programming entropy optimization large and sparse matrices nonorthogonal projections image reconstruction from projections 

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References

  1. 1.
    Robers, P. D., andBen-Israel, A.,An Interval Programming Algorithm for Discrete Linear L 1-Approximation Problems, Journal of Approximation Theory, Vol. 2, pp. 323–336, 1969.Google Scholar
  2. 2.
    Robers, P. D., andBen-Israel, A.,A Suboptimal Method for Interval Linear Programming, Linear Algebra and Its Applications, Vol. 3, pp. 383–405, 1970.Google Scholar
  3. 3.
    Herman, G. T., andLent, A.,A Family of Iterative Quadratic Optimization Algorithms for Pairs of Inequalities, with Application in Diagnostic Radiology, Mathematical Programming Study, Vol. 9, pp. 15–29, 1978.Google Scholar
  4. 4.
    Herman, G. T., andLent, A.,Iterative Reconstruction Algorithms, Computers in Biology and Medicine, Vol. 6, pp. 273–294, 1976.Google Scholar
  5. 5.
    Herman, G. T., Lent, A., andLutz, P. H.,Relaxation Methods For Image Reconstruction, Communications of the Association for Computing Machinery, Vol. 21, pp. 152–158, 1978.Google Scholar
  6. 6.
    Bregman, L. M.,The Relaxation Method of Finding the Common Point of Convex Sets and Its Application to the Solution of Problems in Convex Programming, USSR Computational Mathematics and Mathematical Physics, Vol. 7, pp. 200–217, 1967.Google Scholar
  7. 7.
    Hildreth, C.,A Quadratic Programming Procedure, Naval Research Logistics Quarterly, Vol. 4, pp. 79–85, 1975; see alsoErratum, Naval Research Logistic Quarterly, Vol. 4, p. 361, 1975.Google Scholar
  8. 8.
    Lent, A., andCensor, Y.,Extensions of Hildreth's Row-Action Method for Quadratic Programming, SIAM Journal on Control and Optimization, Vol. 18, pp. 444–454, 1980.Google Scholar
  9. 9.
    D'Esopo, D. A.,A Convex Programming Procedure, Naval Research Logistics Quarterly, Vol. 6, pp. 33–42, 1959.Google Scholar
  10. 10.
    Censor, Y., andHerman, G. T.,Row-Generation Methods for Feasibility and Optimization Problems Involving Sparse Matrices and Their Application, Sparse Matrix Proceedings-1978, Edited by I. S. Duff and G. W. Stewart, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, pp. 197–219, 1979.Google Scholar
  11. 11.
    Censor, Y.,Row-Action Methods for Huge and Sparse Systems and Their Applications, SIAM Review, to appear.Google Scholar
  12. 12.
    Rockafellar, R. T.,Convex Analysis, Princeton University Press, Princeton, New Jersey, 1970.Google Scholar
  13. 13.
    Stoer, J., andWitzgall, C.,Convexity and Optimization in Finite Dimensions, I, Springer-Verlag, Berlin, Germany, 1970.Google Scholar
  14. 14.
    Ponstein, J.,Seven Kinds of Convexity, SIAM Review, Vol. 9, pp. 115–119, 1967.Google Scholar
  15. 15.
    Ben-Israel, A.,Linear Equations and Inequalities on Finite Dimensional, Real or Complex, Vector Spaces: A Unified Theory, Journal of Mathematical Analysis and Applications, Vol. 27, pp. 367–389, 1969.Google Scholar
  16. 16.
    Lent, A.,A Convergent Algorithm for Maximum Entropy Image Restoration, with a Medical X-Ray Application, Image Analysis and Evaluation, Edited by R. Shaw, Society of Photographic Scientists and Engineers, Washington, DC, pp. 249–257, 1977.Google Scholar
  17. 17.
    Daniel, J. W.,The Approximate Minimization of Functionals, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.Google Scholar
  18. 18.
    Censor, Y., Lakshminarayanan, A. V., andLent, A.,Relaxational Methods for Large-Scale Entropy Optimization Problems, with Application in Image Reconstruction, Information Linkage Between Applied Mathematics and Industry, Edited by P. C. C. Wang, Academic Press, New York, New York, pp. 539–546, 1979.Google Scholar

Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

  • Y. Censor
    • 1
  • A. Lent
    • 1
  1. 1.Medical Image Processing Group, Department of Computer ScienceState University of New York at BuffaloAmherst
  2. 2.Department of MathematicsUniversity of Haifa, Mt. CarmelHaifaIsrael
  3. 3.New Ventures, Technicare CorpCleveland

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